# M337: Group theory becomes relevant

One of the things I’ve enjoyed least about my OU Maths journey so far has been group theory.  I ploughed through whole swathes of M208 applying the techniques and not really seeing the relevance1  I found group theory and the proofs related to it tedious.  Mainly because I was proving something that was “obvious”.  However, I’ve always had a healthy acceptance of partial learnings – knowing that if I was being taught a technique then there was a reason for it. Two years later and that reason finally hit me.

I’m working through the first few books of M337 (Complex Analysis)2 and one of the early techniques is the substitution of powers of complex numbers with a more simplified equation:

$z = a + b\textit{i} \\ \mathrm{Solve:} z^4 + z^2 + 8 = 0 \\ \mathrm{using:} w = z^2 \\ w^2 + 4w + 8 = 0$

Which can then be easily solved using the standard quadratic solver to get solutions for $w$ and then resolve back to $x$.  While this is all fairly handle-churning and might seem obvious, this only works because of the proofs of $\mathbb{C}$ being an Abelian group under multiplication and addition and all complex numbers can be treated in the same way as real numbers for algebraic manipulation.  Without these proofs, complex analysis would be a whole lot more frustrating.

All of which is a great reminder that maths is elegant and beautiful and even the bits that you think are dull and pointless suddenly take on a new level of usefulness when you look at them through the lens of far more complex3 problems.

While I may not love group theory, I can now respect it 🙂

1. A curse of pure maths!
3. With and without the pun

Dr Janet is a Molecular Biochemistry graduate from Oxford University with a doctorate in Computational Neuroscience from Sussex. I’m currently studying for a third degree in Mathematics with Open University.

During the day, and sometimes out of hours, I work as a Chief Science Officer. You can read all about that on my LinkedIn page.